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There are two fuses in an electrical device. Let X denote

Statistics for Engineers and Scientists | 4th Edition | ISBN: 9780073401331 | Authors: William Navidi ISBN: 9780073401331 38

Solution for problem 11SE Chapter 2

Statistics for Engineers and Scientists | 4th Edition

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Statistics for Engineers and Scientists | 4th Edition | ISBN: 9780073401331 | Authors: William Navidi

Statistics for Engineers and Scientists | 4th Edition

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Problem 11SE

There are two fuses in an electrical device. Let X denote the lifetime of the first fuse, and let Y denote the lifetime of the second fuse (both in years). Assume the joint probability density function of X and Y is

a. Find P(X ≤ 2 and Y ≤ 3).

b. Find the probability that both fuses last at least 3 years.

c. Find the marginal probability density function of X.

d. Find the marginal probability density function of Y.

e. Are X and Y independent? Explain.

Step-by-Step Solution:

Step 1 of 4 :

 There are two fuses in an electrical device . Let X denote the lifetime of the first fuse, and let Y denote the lifetime of the second fuse (both in years). And the joint probability density function of X and Y is

 

We have to find

 P(X ≤ 2 and Y ≤ 3).The probability that both fuses last at least 3 years.  The marginal probability density function of X.  The marginal probability density function of Y.  Check X and Y are independent.

Step 2 of 4:

Here X denote the lifetime of first fuse. And Y denote the lifetime of second fuse.

         

So to find  P(X ≤ 2 and Y ≤ 3) we have to integrate the  joint density function in the given     range.  

         

                P(X ≤ 2 and Y ≤ 3)  = 

             

                                           

                                              =   

                           

                                           =    1/6  [ / -1/3  dx

                                             =     - 1/2  [ [] dx

                           

                                       

                                             =       (-1

                                             =      0.3995.

(b)  The probability that both fuses last at least 3 years = P(X, Y)

                                                         

                                  P(, )  =   f(x,y) dx dy

 

                                                           =      

                                                   

           

                                                           =   -1/2  dx

 

                                                       

                                                            =   [

 

                                                            =  

                                                            =  0.0821.

Step 3 of 4

Chapter 2, Problem 11SE is Solved
Step 4 of 4

Textbook: Statistics for Engineers and Scientists
Edition: 4
Author: William Navidi
ISBN: 9780073401331

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